Ron Barnette's Zeno's Coffeehouse Challenge #54 Result

This challenge prompted a rich diversity of replies, and raised many issues that were perceived as relevant to Zeno's Challenge #54. Over 200 submissions were received and appreciated! I have listed below the original Challenge, followed by a sampling of replies and comments. I want to thank ALL respondents for their thoughtful time taken with Zeno's Coffeehouse, and encourage your continued support, as critical thinking exercises are explored for mental growth and recreation.

Thanks!...Ron Barnette

Infinite Beliefs?
While discussing the concepts of infinity and belief late one night at the Coffeehouse, our man Charles was reminded of a wonderful book written back in 1962 by the philosopher Jaakko Hintikka: Knowledge and Belief, in which he develops a working logic of these two notions. Among the semantics developed and argued is the plausible conditional 'If one believes some proposition, p, then one believes that one believes that p.' The seemingly obvious idea is that one can't believe that something is the case without at least believing that one believes it. 'Charles believes that Maggie is at the store, but Charles doesn't believe that he believes this' does appear to be plainly false, to be sure, if not bordering on an inconsistency. Letting the Hintikka logical notation 'Bap' be read as 'a believes that p,' the above semantic implication is, thus:
1. Bap > BaBap
So far, so good, you say. Upon hearing this, dear Maggie (who's not at the store, obviously!) jumped into the Coffeehouse discussion and added a Zenoesque comment and challenge:
"Hold on, Charles. If we let 'Bap' be the proposition believed by a above in the consequent condition, then, according to 1, we can derive that
2. BaBap > BaBaBap,
and hence that
3. BaBaBap > BaBaBaBap
and so on ad infinitum! So what you and Hintikka are saying is that if one has any belief at all, then one has an infinite number of beliefs? Charles, this sounds crazy! Am I missing something? What sounded like an obvious truism seems to entail that believers are like Zeno's racers...you can't move unless you move forever! Yikes!"
Is Maggie correct? Does the maxim that one believes what one believes entail that one has an infinite number of beliefs? Is this a problem? Help!

RESPONSES: In addition to the many, many submissions who agreed with Maggie, many others offered varieties of accounts which dealt with associated matters I list below with sample replies. I hope you enjoy the entries listed for further reflection and critical thought..RB

1. Re: Infinite beliefs and meaningfulness:

Troy Williamson:

Thanks for the accolades, Ron, although I have to admit that succinctness is not always a virtue that I display publicly!  ;-)

As for the troubling aspect of this, I'm not sure that we should be bothered too much--when we realize that the implications we're talking about don't really have "real world" equivalencies.  We might liken it to mathematics--take the most complicated expression you want, raise the whole thing to the 0 power, and the result is 1.  What does that mean?  It means that the value "1" can be written in really complicated, convoluted ways that have no real meaning (if all you are trying to do is express the value "1").  It is interesting sometimes--but not necessarily troubling.

Admitted, there IS a troubling aspect if meaning is lost.  I'm still not sure what someone would mean, though, if they say, "I believe that I believe that I believe that p."  It seems to me that the meaning has already been lost before we even begin to analyze the logical "value" of the statement.  But maybe I'm overlooking the obvious (??).
Troy

> ----- Original Message -----
> From: "Ron Barnette" <rbarnett@valdosta.edu>
> To: "Troy Williamson"
> Subject: Re: Submitted Form
> Date: Fri, 19 May 2006 12:27:12 -0400
> Troy,
> What a wonderful set of comments on the latest Coffeehouse
> challenge! Thanks for your lucid explication---not unlike my own, I
> hasten to add! (Only yours is much more succint, ie., elegant:)
> Cheers,
> Ron
> PS It does trouble me a bit, though, that we're left with the claim
> that at least some logical implications (or virtual implications)
> are unintelligible! I would think that implicature should preserve
> meaningfulness:)
> RB
> ----- Original Message ----- From: "Troy Williamson"

> > Troy Williamson
> > city: Abilene
> > state: TX country: USA
> > comments:      The problem, simply stated, is this:   If Bap
> > implies BaBap, then that implies BaBaBap, which implies
> > BaBaBaBap, and so on.  But is it intelligible and/or acceptable
> > for these implications to exist?
> >     The answer is, "Yes and no."
> >     First, notice that the implication referred to above is a
> > "virtual implication," in Hintikka's own words (p. 123).  That
> > is, when Bap is true, then BaBap will also be true (logically
> > speaking).  However, Hintikka does point out that this virtual
> > implication only works one direction; BaBap does not logically
> > imply Bap.
> >     Because of this, we can say that the infinite chain of
> > beliefs is a byproduct of logic.  There is no problem with such a
> > chain evolving-it is a natural occurrence.  The question of
> > acceptability within the logical system is a moot
> > point-equivalent to asking whether it is acceptable for p to
> > imply (p & p).  The infinite chain of iterations, logically
> > speaking, is unavoidable.
> >     However, the question of the intelligibility of such an
> > iteration is different.  In his discussion of "knowing that one
> > knows," Hintikka points out that quite often the logical
> > implication is not all that is involved in such a statement.  If
> > someone says, "I know that I know p," then they seem to be
> > telling us more than a fact about their knowledge.  The speaker
> > may be referring to a certain awareness of the knowledge he
> > possesses; the speaker may be seeking to emphasize the certainty
> > of his knowledge about p.
> >     The same seems to apply to a belief about a belief.   To
> > believe p means, roughly, that the individual in question holds a
> > certain conviction regarding the truth of p.  To say that the
> > person believes that he believes that p, however, refers to a
> > different type of belief; there is an added connotation involved.
> >  In this particular case (as opposed to knowing that one knows),
> > it seems that the statement grows weaker.  The speaker seems to
> > be suggesting a note of doubt-"I believe that my belief is p, but
> > perhaps I am mistaken about my own belief."  (Since belief is a
> > conviction that has less substantiation than knowledge, to
> > believe in reference to another belief seems to weaken the claim.)
> > wholly unclear.  When one says, "I believe that I believe that I
> > believe that p," for example, it is not at all obvious what the
> > additional iteration implies-in essence, it becomes nonsensical.
> > And this would hold true as additional iterations are added to
> > the statement.  Indeed, this appears to be one of those
> > situations where we can clearly discuss the logical meaning of a
> > statement which would never occur in real life-because the
> > statement is so convoluted as to mask any intelligible meaning
> > that it might seek to convey.
> >     So is it acceptable for this infinite chain of beliefs to
> > exist?  Yes, in the sense that there is a virtual implication
> > which will naturally arise within a logical system.  Is it
> > intelligible for this infinite chain of beliefs to exist?  Such
> > statements are not common in the "everyday world."  When they do
> > occur, their intelligibility suffers greatly as the iterations
> > increase-seemingly uttered for some purpose whose intent has been
> > lost.  Becoming nonsensical, there is no intelligibility to the
> > longer phrases.  But since they are seldom, if ever, encountered
> > outside of a logician's musings, does it really matter?
> >     I believe that this is my answer.  And I believe that I
> > believe that this is my answer.  But if I say more, my statement
> > will be reduced to a nonsensical utterance, so I will just stop
> > right

2. Re: Challenges to Hintikka:

John Cogan: city: Clearwater

state: Fl

country: usa

comments: I respectfully disagree with Prof. Hintikka's conclusion that if one believes p that, therefore, one also believes that one believes p on the grounds that the epistemic relationship between me and my beliefs is not the same as that between me and p. The epistemic relationship obtaining between me an my beliefs is not one of belief but of knowledge because no grounds could be mustered to support  doubt that I believe that p; hence, with no possibility of doubt, the epistemic relation must be knowledge.

Bob MacIntosh: city: Llandudno

state: Wales

country: UK

comments: The regress doesn't seem too vicious as long as belief does not require actually doing or thinking anything; a definition involving some sort of  'if asked, will be inclined to admit.' However, I might wish to say of someone, 'he believes that he believes that all men are created equal in dignity and rights, but I can see from his actions that he believes the contrary.' (no names, no impeachment). Our capacity for self-deception seems to be unfettered by considerations of logic and consistency, so I believe that the original proposition is wrong - at least that's what I believe I believe, until someone convinces me otherwise.

Larry Colter: city: Evansville

state: IN

country: USA

comments: First, a clarification. Your penultimate question above is not equivalent to Hintikka's "Bap > BaBap". Your question has the trivial answer "yes". Hintikka's proposition, however, seems just false, at least with respect to humans; for we frequently discover that we (dispositionally) believe a proposition that we didn't know (or believe) that we believed. If H's proposition not false in that way, however, perhaps it's false in this way: we don't have the iteratin of beliefs that H requires, but instead the "believing that we believe" is built directly into Bap. Thus, perhaps, Bap is an inadequate way of expressing what is going on. Perhaps the relevant wording is something like "I believe that p and part of what I believe or am aware of is that I do believe that p".

Isaac Creek: city: Evansville

state: IN

country: USA

comments: First, a clarification. Your penultimate question above is not equivalent to Hintikka's "Bap > BaBap". Your question has the trivial answer "yes". Hintikka's proposition, however, seems just false, at least with respect to humans; for we frequently discover that we (dispositionally) believe a proposition that we didn't know (or believe) that we believed. If H's proposition not false in that way, however, perhaps it's false in this way: we don't have the iteratin of beliefs that H requires, but instead the "believing that we believe" is built directly into Bap. Thus, perhaps, Bap is an inadequate way of expressing what is going on. Perhaps the relevant wording is something like "I believe that p and part of what I believe or am aware of is that I do believe that p".

3. Re: Consciousness and beliefs; 1st & 2nd order beliefs; occurrent/tacit beliefs; time & phenomenology & beliefs:

Liz McKinnell: city: Durham

country: UK

comments: It could be argued that Bap>BaBap is mistaken.  It seems obvious, because when we think about something we believe, OF COURSE we believe that we believe it.   If we didn't, we wouldn't have thought about it as something that we believe.   There are two possible explanations of this:

1)  We can subconciously believe things that we do not believe that we believe (but false consciousness arguments are a bit ropey)

or

2) Bap>BaBap applies to APPARENT beliefs- thoughts rather than states of belief.   It is only true that 'I believe that p' entails 'I believe that I believe that p' when I actually think about my belief that p.  'I believe that I believe thatp' is not often an apparent belief (unless you are doing this sort of puzzle), so it needn't enatail 'I believe that I believe that I believe that p'.  We would not have the time or mental capacity (let alone the patience) to conceptualise a long chain, never mind an infinite one, of belief entailments of this sort.

J. S. Green: city: London

country: England

comments: May I (v. briefly)suggest that the concept of 'belief' is quite ambiguous and it is the ambiguity that leads to the infinit regress (so there isn't really any logical problem; what is needed is rather an epistemological clarification of the concept of belief).  Maggie's second-order 'belief' that she has a first-order belief is of a different nature from her first-order belief (hence it is to equivocate to say that Bap > BaBap). MAggie beliefs p in a first order manner and is AWARE that she believes that p. It is the awareness of her believing that p which, fallaciously, gets formulated as BaBap.

Leon: city: Melbourne

country: Australia

comments: The idea that if we believe p, then we must believe that we believe it is quite plausible upon intuitive reflection, but Maggie is quite right that it is undefendable on logical grounds. What we need is an explanation that accounts for our intuitions, but doesn't run into this logical problem. I suggest the following.

There are two types of beliefs: occurrent beliefs and tacit beliefs. Occurrent beliefs are beliefs whose content we are currently attending to or thinking. For instance, Charles might right now be attending to the fact that Maggie is in the store, and hence he believes it. Tacit beliefs are beliefs whose content we're not currently attending to, but an appropriate stimuli might easily prompt a shift of attention, and hence make a tacit belief occurrent. Now, when one has an occurrent belief that p, one necessarily has a tacit belief that one believes that p. But when one has a tacit belief there is no further belief about that belief.

This explains our intuition because a tacit belief about an occurrent belief can easily become active because the occurrent belief is available to consciousness. All that's needed is introspection. Consequently, whenever we have an occurrent belief that p, it is no surprise that we think we also believe that we believe that p. Whenever a tacit belief becomes occurrent it takes almost no effort to make that belief the focus of a further belief, and for that further belief the focus of another belief, and so on. There is a tacit belief waiting right around the introspective corner!

The upshot is this. Psychologically, it really does feel like there is an infinite regress of beliefs about beliefs, but this is merely an illusion. Let B = occurrent belief and B* = tacit belief. The implication now becomes this:

1. Bap > B*ap

But it is not the case that B*ap > B*ap. There is no logical regress, only an indefinite psychological reflectiveness.

country: UK

comments: Maggie is completely correct in her claim that the maxim leads to fatal regress.   The problem can be solved quite simply by realising that belief is a present-tensed phenomenon.  One can only ever be said to 'have' a belief at the time at which one is contemplating it.  It follows from this that unless Charles is contemplating his belief in his belief he simply doesn't have it.  It is important when considering this to recognise that vast array of automatic habituated responses which we have to the world which are often designated 'beliefs' e.g. my belief that the water in the glass will be thirst-quenching.  At an everyday level of description I canbe said to hold this belief every time I drink, however since 'belief' should is a mental term and only experiential goings-on can ever be termed mental, and further since present experiential goings-on are the only ones that can go on, we only hold beliefs when they factor consicously in our experience.

Nollaig MacKensie: Yes, I meant '~' as 'it is not the case that'. So
'Charles believes that Maggie is at the store', would,
according to my suggestion, imply

'Its not the case that Charles believes that its not
the case that Charles believes that Maggie is
at the store'

This is, of course, weaker than 'Charles believes that
Charles believes....'.

So rather than saying there's something amiss with

'Charles believes that Maggie is at the store,
but Charles doesn't believe that he believes this',

'Charles believes that Maggie is at the store, but
Charles believes that he doesn't believe this'.

(The weaker line goes very nicely if you replace 'believe'
with 'assert')

(Someone who took an extreme dispositionalist line on
belief, say 'Bap <=> a is disposed to assent to p',
might not be troubled by infinite beliefs. Charles
might be disposed to assent to 'Charles is disposed to
assent to "Charles is disposed to assent to... <close
layers of quotes>, if anyone ever raised the question :-)

......N.

On 2006.05.22 19:11:48, you,
the extraordinary Ron Barnette, opined:

Nollaig
> Thanks for your observation and visit to the Coffeehouse. I do have a
> question, though: are the negations of B ('believes that') in your
> entailment conseqent meant as 'a disbelieves that' or simply as a denial
> that a believes that x is the case? These are different, obviously. My
> comment on the challenge was meant as the latter, if you read me right as
> of this as a 'virtual' implication, so as not to confuse the matter.
> But aside from this caveat, it seems to me that if one believes that
> something is the case, then one does believe that one believes it, no? The
> problem arises somewhere during the infinite reiteration, however, and that
> is what the challenge is aimed at exploring.
> Best,
> Ron

> ----- Original Message -----
> From: "Nollaig MacKenzie"
> >city: Toronto
> >state: ON
> >'Charles believes that Maggie is at the store, but Charles doesn't believe
> >that he believes this' does appear to be plainly false, to be sure, if not
> >bordering on an inconsistency."
> >is too strong, I think. My shoot-from-the-hip counter-intuition is:
> >if a is a rational person,
> >Bap => ~Ba(~Bap)

Ben Howard-McKinney:

From: "Ron Barnette" <rbarnett@valdosta.edu>
To: "Ben Howard-McKinney"

I appreciate your Zeno's response, Ben. Thanks for taking the time. I do
have one question, though, related to your notion of 'one's realizing that.'
You tacitly imply that an infinite number of such realizations WOULD BE a
problem, unlike that of an infinite number of beliefs, of which you don't
see a problem, as you state. Why the difference? Isn't realizing that
something is the case tantamount to saying that one has a true belief that
something is the case? Or does realizing mean something more for you, in
that to realize something implies that one 'holds the belief in one's
present conscious state,' or something to that effect.
At any rate, I wanted to drop you a note of thanks.
Best,
Ron Barnette
----- Original Message -----
From: "Ben Howard-McKinney"
To: <rbarnett@valdosta.edu>

> Below is the result of your feedback form. It was submitted by
> Ben Howard-McKinney

> ---------------------------------------------------------------------------
>
> city: Normal
>
> state: IL
>
> comments: I don't see any problem with being able to hold an infinite
> number of beliefs. It is not the case that a person must constantly be
> realizing that they believe that they believe. It is just that an infinite
> number of beliefs are derivable from one. By comparison, an infinite
> number of mathematical truths can be derived from the realization that 1
> and 1 make 2, such as the fact that 2 minus 1 is 1, and that 1 and 1 and 1
> and 1 are 4. In sentential logic, from the atomic sentence A, we can also
> come up with infinite derivations, such as A & A , A v B , A & (B v ~ B),
> and so on.